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The Learning-With-Errors (LWE) problem is a crucial computational challenge with significant implications for post-quantum cryptography and computational learning theory. Here we propose a quantum-classical hybrid algorithm with Ising model (HAWI) to address the LWE problem. Our approach involves transforming the LWE problem into the Shortest Vector Problem (SVP), using variable qubits to encode lattice vectors into an Ising Hamiltonian. We then identify the low-energy levels of the Hamiltonian to extract the solution, making it suitable for implementation on current noisy intermediate-scale quantum (NISQ) devices. We prove that the number of qubits required is less than $m(3m-1)/2$, where $m$ is the number of samples in the algorithm. Our algorithm is heuristic, and its time complexity depends on the specific quantum algorithm employed to find the Hamiltonian's low-energy levels. If the Quantum Approximate Optimization Algorithm (QAOA) is used to solve the Ising Hamiltonian problem, and the number of iterations satisfies $y < O\left(m\log m\cdot 2^{0.2972k}/pk^2\right)$, our algorithm will outperform the classical Block Korkine-Zolotarev (BKZ) algorithm, where $k$ is the block size related to problem parameters, and $p$ is the number of layers in QAOA. We demonstrate the algorithm by solving a $2$-dimensional LWE problem on a real quantum device with $5$ qubits, showing its potential for solving meaningful instances of the LWE problem in the NISQ era.

We introduce MindSpore Quantum, a pioneering hybrid quantum-classical framework with a primary focus on the design and implementation of noisy intermediate-scale quantum (NISQ) algorithms. Leveraging the robust support of MindSpore, an advanced open-source deep learning training/inference framework, MindSpore Quantum exhibits exceptional efficiency in the design and training of variational quantum algorithms on both CPU and GPU platforms, delivering remarkable performance. Furthermore, this framework places a strong emphasis on enhancing the operational efficiency of quantum algorithms when executed on real quantum hardware. This encompasses the development of algorithms for quantum circuit compilation and qubit mapping, crucial components for achieving optimal performance on quantum processors. In addition to the core framework, we introduce QuPack, a meticulously crafted quantum computing acceleration engine. QuPack significantly accelerates the simulation speed of MindSpore Quantum, particularly in variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and tensor network simulations, providing astonishing speed. This combination of cutting-edge technologies empowers researchers and practitioners to explore the frontiers of quantum computing with unprecedented efficiency and performance.

Quantum computing is a promising candidate for accelerating machine learning tasks. Limited by the control accuracy of current quantum hardware, reducing the consumption of quantum resources is the key to achieving quantum advantage. Here, we propose a quantum resonant dimension reduction (QRDR) algorithm based on the quantum resonant transition to reduce the dimension of input data and accelerate the quantum machine learning algorithms. After QRDR, the dimension of input data $N$ can be reduced into desired scale $R$, and the effective information of the original data will be preserved correspondingly, which will reduce the computational complexity of subsequent quantum machine learning algorithms or quantum storage. QRDR operates with polylogarithmic time complexity and reduces the error dependency from the order of $1/\epsilon^3$ to the order of $1/\epsilon$, compared to existing algorithms. We demonstrate the performance of our algorithm combining with two types of quantum classifiers, quantum support vector machines and quantum convolutional neural networks, for classifying underwater detection targets and quantum many-body phase respectively. The simulation results indicate that reduced data improved the processing efficiency and accuracy following the application of QRDR. As quantum machine learning continues to advance, our algorithm has the potential to be utilized in a variety of computing fields.

Simulating quantum physical processes has been one of the major motivations for quantum information science. Quantum channels, which are completely positive and trace preserving processes, are the standard mathematical language to describe quantum evolution, while in recent years quantum superchannels have emerged as the substantial extension. Superchannels capture effects of quantum memory and non-Markovianality more precisely, and have found broad applications in universal models, algorithm, metrology, discrimination tasks, as examples. Here, we report an experimental simulation of qubit superchannels in a nuclear magnetic resonance (NMR) system with high accuracy, based on a recently developed quantum algorithm for superchannel simulation. Our algorithm applies to arbitrary target superchannels, and our experiment shows the high quality of NMR simulators for near-term usage. Our approach can also be adapted to other experimental systems and demonstrates prospects for more applications of superchannels.

There has been an increasing research focus on quantum algorithms for condensed matter systems recently, particularly on calculating energy band structures. Here, we propose a quantum algorithm, the powered full quantum eigensolver(P-FQE), by using the exponentiation of operators of the full quantum eigensolver(FQE). This leads to an exponential increase in the success probability of measuring the target state in certain circumstances where the number of generating elements involved in the exponentiation of operators exhibit a log polynomial dependence on the number of orbitals. Furthermore, we conduct numerical calculations for band structure determination of the twisted double-layer graphene. We experimentally demonstrate the feasibility and robustness of the P-FQE algorithm using superconducting quantum computers for graphene and Weyl semimetal. One significant advantage of our algorithm is its ability to reduce the requirements of extremely high-performance hardware, making it more suitable for energy spectra determination on noisy intermediate-scale quantum (NISQ) devices.

Shor's algorithm has seriously challenged information security based on public key cryptosystems. However, to break the widely used RSA-2048 scheme, one needs millions of physical qubits, which is far beyond current technical capabilities. Here, we report a universal quantum algorithm for integer factorization by combining the classical lattice reduction with a quantum approximate optimization algorithm (QAOA). The number of qubits required is O(logN/loglog N), which is sublinear in the bit length of the integer $N$, making it the most qubit-saving factorization algorithm to date. We demonstrate the algorithm experimentally by factoring integers up to 48 bits with 10 superconducting qubits, the largest integer factored on a quantum device. We estimate that a quantum circuit with 372 physical qubits and a depth of thousands is necessary to challenge RSA-2048 using our algorithm. Our study shows great promise in expediting the application of current noisy quantum computers, and paves the way to factor large integers of realistic cryptographic significance., Comment: 32 pages, 12 figures

We propose a variational quantum attack algorithm (VQAA) for classical AES-like symmetric cryptography, as exemplified the simplified-data encryption standard (S-DES). In the VQAA, the known ciphertext is encoded as the ground state of a Hamiltonian that is constructed through a regular graph, and the ground state can be found using a variational approach. We designed the ansatz and cost function for the S-DES's variational quantum attack. It is surprising that sometimes the VQAA is even faster than Grove's algorithm as demonstrated by our simulation results. The relationships of the entanglement entropy, concurrence and the cost function are investigated, which indicate that entanglement plays a crucial role in the speedup.

Utilizing quantum computer to investigate quantum chemistry is an important research field nowadays. In addition to the ground-state problems that have been widely studied, the determination of excited-states plays a crucial role in the prediction and modeling of chemical reactions and other physical processes. Here, we propose a non-variational full circuit-based quantum algorithm for obtaining the excited-state spectrum of a quantum chemistry Hamiltonian. Compared with previous classical-quantum hybrid variational algorithms, our method eliminates the classical optimization process, reduces the resource cost caused by the interaction between different systems, and achieves faster convergence rate and stronger robustness against noise without barren plateau. The parameter updating for determining the next energy-level is naturally dependent on the energy measurement outputs of the previous energy-level and can be realized by only modifying the state preparation process of ancillary system, introducing little additional resource overhead. Numerical simulations of the algorithm with hydrogen, LiH, H2O and NH3 molecules are presented. Furthermore, we offer an experimental demonstration of the algorithm on a superconducting quantum computing platform, and the results show a good agreement with theoretical expectations. The algorithm can be widely applied to various Hamiltonian spectrum determination problems on the fault-tolerant quantum computers., Comment: accepted for publication in Quantum

Advanced Encryption Standard(AES) is one of the most widely used block ciphers nowadays, and has been established as an encryption standard in 2001. Here we design AES-128 and the sample-AES(S-AES) quantum circuits for deciphering. In the quantum circuit of AES-128, we perform an affine transformation for the SubBytes part to solve the problem that the initial state of the output qubits in SubBytes is not the $\ket{0}^{\otimes 8}$ state. After that, we are able to encode the new round sub-key on the qubits encoding the previous round sub-key, and this improvement reduces the number of qubits used by 224 compared with Langenberg et al.'s implementation. For S-AES, a complete quantum circuit is presented with only 48 qubits, which is already within the reach of existing noisy intermediate-scale quantum computers.

Quantum machine learning is one of the most promising applications of quantum computing in the Noisy Intermediate-Scale Quantum(NISQ) era. Here we propose a quantum convolutional neural network(QCNN) inspired by convolutional neural networks(CNN), which greatly reduces the computing complexity compared with its classical counterparts, with $O((log_{2}M)^6) $ basic gates and $O(m^2+e)$ variational parameters, where $M$ is the input data size, $m$ is the filter mask size and $e$ is the number of parameters in a Hamiltonian. Our model is robust to certain noise for image recognition tasks and the parameters are independent on the input sizes, making it friendly to near-term quantum devices. We demonstrate QCNN with two explicit examples. First, QCNN is applied to image processing and numerical simulation of three types of spatial filtering, image smoothing, sharpening, and edge detection are performed. Secondly, we demonstrate QCNN in recognizing image, namely, the recognition of handwritten numbers. Compared with previous work, this machine learning model can provide implementable quantum circuits that accurately corresponds to a specific classical convolutional kernel. It provides an efficient avenue to transform CNN to QCNN directly and opens up the prospect of exploiting quantum power to process information in the era of big data., Comment: we need a major revision for our manuscript and add some new contexts

Gradient-based algorithms, popular strategies to optimization problems, are essential for many modern machine-learning techniques. Theoretically, extreme points of certain cost functions can be found iteratively along the directions of the gradient. The time required to calculating the gradient of $d$-dimensional problems is at a level of $\mathcal{O}(poly(d))$, which could be boosted by quantum techniques, benefiting the high-dimensional data processing, especially the modern machine-learning engineering with the number of optimized parameters being in billions. Here, we propose a quantum gradient algorithm for optimizing general polynomials with the dressed amplitude encoding, aiming at solving fast-convergence polynomials problems within both time and memory consumption in $\mathcal{O}(poly (\log{d}))$. Furthermore, numerical simulations are carried out to inspect the performance of this protocol by considering the noises or perturbations from initialization, operation and truncation. For the potential values in high-dimension optimizations, this quantum gradient algorithm is supposed to facilitate the polynomial-optimizations, being a subroutine for future practical quantum computer., Comment: (5+10)pages + (3+1) figures. Comments are always welcome

Finding a maximum or minimum is a fundamental building block in many mathematical models. Compared with classical algorithms, Durr, Hoyer's quantum algorithm (DHA) achieves quadratic speed. However, its key step, the quantum exponential searching algorithm (QESA), which is based on Grover algorithm, is not a sure-success algorithm. Meanwhile, quantum circuits encounter the gate decomposition problem due to variation of the scale of data. In this paper, we propose an optimized quantum algorithm for searching maximum and minimum, based on DHA and the optimal quantum exact search algorithm. Furthermore, we provide the corresponding quantum circuits, together with three equivalent simplifications. In circumstances when we can exactly estimate the ratio of the number of solutions M and the searched space N, our method can improve the successful probability close to 100%. Furthermore, compared with DHA, our algorithm shows an advantage in complexity with large databases and in the gate complexity of constructing oracles. Experiments have been executed on an IBM superconducting processor with two qubits, and a practical problem of finding the minimum from Titanic passengers' age was numerically simulated. Both showed that our optimized maximum or minimum performs more efficiently compared with DHA. Our algorithm can serve as an important subroutine in various quantum algorithms which involves searching maximum or minimum.

Quantum simulation of quantum chemistry is one of the most compelling applications of quantum computing. It is of particular importance in areas ranging from materials science, biochemistry and condensed matter physics. Here, we propose a full quantum eigensolver (FQE) algorithm to calculate the molecular ground energies and electronic structures using quantum gradient descent. Compared to existing classical-quantum hybrid methods such as variational quantum eigensolver (VQE), our method removes the classical optimizer and performs all the calculations on a quantum computer with faster convergence. The gradient descent iteration depth has a favorable complexity that is logarithmically dependent on the system size and inverse of the precision. Moreover, the FQE can be further simplified by exploiting perturbation theory for the calculations of intermediate matrix elements, and obtain results with a precision that satisfies the requirement of chemistry application. The full quantum eigensolver can be implemented on a near-term quantum computer. With the rapid development of quantum computing hardware, FQE provides an efficient and powerful tool to solve quantum chemistry problems.

The recently theoretical and experimental researches related to $\mathcal{PT}$-symmetric system have attracted unprecedented attention because of various novel features and potentials in extending canonical quantum mechanics. However, as the counterpart of $\mathcal{PT}$-symmetry, there are only a few researches on anti-$\mathcal{PT}$-symmetry. Here, we propose an algorithm for simulating the universal anti-$\mathcal{PT}$-symmetric system with quantum circuit. Utilizing the protocols, an oscillation of information flow is observed for the first time in our Nuclear Magnetic Resonance quantum simulator. We will show that information will recover from the environment completely when the anti-$\mathcal{PT}$-symmetry is broken, whereas no information can be retrieved in the symmetry-unbroken phase. Our work opens the gate for practical quantum simulation and experimental investigation of universal anti-$\mathcal{PT}$-symmetric system in quantum computer.

The ability of implementing quantum operations plays fundamental role in manipulating quantum systems. Creation and annihilation operators which transform a quantum state to another by adding or subtracting a particle are crucial of constructing quantum description of many body quantum theory and quantum field theory. Here we present a quantum algorithm to perform them by the linear combination of unitary operations associated with a two-qubit ancillary system. Our method can realize creation and annihilation operators simultaneously in the subspace of the whole system. A prototypical experiment was performed with a 4-qubit Nuclear Magnetic Resonance processor, demonstrating the algorithm via full state tomography. The creation and annihilation operators are realized with a fidelity all above 96% and a probability about 50%. Moreover, our method can be employed to quantum random walk in an arbitrary initial state. With the prosperous development of quantum computing, our work provides a quantum control technology to implement non-unitary evolution in near-term quantum computer., Comment: 7 pages, 8 figures

We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an $N\times N$ matrix $\mathcal{M}$, an $N$-dimensional vector $\textbf{\emph{b}}$, and an initial vector $\textbf{\emph{x}}(0)$, obtain a target vector $\textbf{\emph{x}}(t)$ as a function of time $t$ according to the constraint $d\textbf{\emph{x}}(t)/dt=\mathcal{M}\textbf{\emph{x}}(t)+\textbf{\emph{b}}$. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances. In addition, we demonstrate our quantum algorithm for a $4\times4$ linear differential equation using a 4-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations., Comment: 12 pages, 5 figures, and all comments are welcome!

Quantum adiabatic algorithm is of vital importance in quantum computation field. It offers us an alternative approach to manipulate the system instead of quantum gate model. Recently, an interesting work arXiv:1805.10549 indicated that we can solve linear equation system via algorithm inspired by adiabatic quantum computing. Here we demonstrate the algorithm and realize the solution of 8-dimensional linear equations $A\textbf{x}=\textbf{b}$ in a 4-qubit nuclear magnetic resonance system. The result is by far the solution of maximum-dimensional linear equation with a limited number of qubits in experiments, which includes some ingenious simplifications. Our experiment provides the new possibility of solving so many practical problems related to linear equations systems and has the potential applications in designing the future quantum algorithms.

Gradient descent method, as one of the major methods in numerical optimization, is the key ingredient in many machine learning algorithms. As one of the most fundamental way to solve the optimization problems, it promises the function value to move along the direction of steepest descent. For the vast resource consumption when dealing with high-dimensional problems, a quantum version of this iterative optimization algorithm has been proposed recently[arXiv:1612.01789]. Here, we develop this protocol and implement it on a quantum simulator with limited resource. Moreover, a prototypical experiment was shown with a 4-qubit Nuclear Magnetic Resonance quantum processor, demonstrating a optimization process of polynomial function iteratively. In each iteration, we achieved an average fidelity of 94\% compared with theoretical calculation via full-state tomography. In particular, the iterative point gradually converged to the local minimum. We apply our method to multidimensional scaling problem, further showing the potentially capability to yields an exponentially improvement compared with classical counterparts. With the onrushing tendency of quantum information, our work could provide a subroutine for the application of future practical quantum computers., Comment: 6+4 pages, 8 figures

The quantum clock synchronization (QCS) is to measure the time difference among the spatially separated clocks with the principle of quantum mechanics. The first QCS algorithm proposed by Chuang and Jozsa is merely based on two parties, which is further extended and generalized to the multiparty situation by Krco and Paul. They present a multiparty QCS protocol based upon W states that utilizes shared prior entanglement and broadcast of classical information to synchronize spatially separated clocks. Shortly afterwards, Ben-Av and Exman came up with an optimized multiparty QCS using Z state. In this work, we firstly report an implementation of Krco and Ben-AV multiparty QCS algorithm using a four-qubit Nuclear Magnetic Resonance (NMR). The experimental results show a great agreement with the theory and also prove Ben-AV multiparty QCS algorithm more accurate than Krco., Comment: 6 pages and 6 figures. All comments are welcome!